Who are the Big-Endians?

Allegorically, the Big-endians represent Roman Catholics and the Little-endians represent Protestants, and the war between them is representative of the religious struggles and persecution of these groups at various times in British history. Essentially, in the text, the Big-endians want to break their eggs at the bigger end, and the Little-endians want to break their eggs at the smaller end. The quarrel is, of course, ridiculous, because, no matter which side you crack, you get to the egg inside. The difference between these groups of people is minuscule, momentary only. However, they have fought and died for their causes regardless. Gulliver tells us,

[...] the words are these: ‘that all true believers break their eggs at the convenient end.’ And which is the convenient end, seems, in my humble opinion to be left to every man’s conscience, or at least in the power of the chief magistrate to determine. Now, the Big-endian exiles have found so much credit in the emperor of Blefuscu’s court, and so much private assistance and encouragement from their party here at home, that a bloody war has been carried on between the two empires for six-and-thirty moons [...].

Swift suggests, then, that the difference between Catholics and Protestants is likewise inconsequential: if Catholics want to believe in miracles while Protestants do not, who cares? The actions and beliefs of one group need not affect the actions and beliefs of the other. However, people fought and died for years as a result of this clash of religious ideas, and Swift evidently believes that such violence and loss of life was ridiculous, given the cause.

The Big-Endians are a political faction who stage a rebellion against the Lilliputian Emperor. The Emperor decrees that when eating a hard-boiled egg, it should always be broken at the little end. The Big-Endians, on the other hand, prefer to break their eggs at the big end, which is looked down upon by the Emperor as a primitive custom. The Emperor's great-grandfather instituted the Little-Endian custom after his son cut his finger trying to break a boiled egg at the big end.
This seemingly trivial argument develops into a full-scale civil war between the two rival factions. Swift is satirizing the ceaseless religious conflicts that had plagued England ever since the Reformation. Big-Endians represent Roman Catholics, and Little-Endians are Protestants.